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Lyubetsky et al. Biology Direct 2012, 7:26
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RESEARCH
Open Access
Modeling RNA polymerase interaction in
mitochondria of chordates
Vassily A Lyubetsky*, Oleg A Zverkov, Sergey A Pirogov, Lev I Rubanov and Alexandr V Seliverstov
Abstract
Background: In previous work, we introduced a concept, a mathematical model and its computer realization that
describe the interaction between bacterial and phage type RNA polymerases, protein factors, DNA and RNA
secondary structures during transcription, including transcription initiation and termination. The model accurately
reproduces changes of gene transcription level observed in polymerase sigma-subunit knockout and heat shock
experiments in plant plastids. The corresponding computer program and a user guide are available at http://lab6.
iitp.ru/en/rivals. Here we apply the model to the analysis of transcription and (partially) translation processes in the
mitochondria of frog, rat and human. Notably, mitochondria possess only phage-type polymerases. We consider
the entire mitochondrial genome so that our model allows RNA polymerases to complete more than one circle on
the DNA strand.
Results: Our model of RNA polymerase interaction during transcription initiation and elongation accurately
reproduces experimental data obtained for plastids. Moreover, it also reproduces evidence on bulk RNA
concentrations and RNA half-lives in the mitochondria of frog, human with or without the MELAS mutation, and rat
with normal (euthyroid) or hyposecretion of thyroid hormone (hypothyroid). The transcription characteristics
predicted by the model include: (i) the fraction of polymerases terminating at a protein-dependent terminator in
both directions (the terminator polarization), (ii) the binding intensities of the regulatory protein factor (mTERF) with
the termination site and, (iii) the transcription initiation intensities (initiation frequencies) of all promoters in all five
conditions (frog, healthy human, human with MELAS syndrome, healthy rat, and hypothyroid rat with aberrant
mtDNA methylation). Using the model, absolute levels of all gene transcription can be inferred from an arbitrary
array of the three transcription characteristics, whereas, for selected genes only relative RNA concentrations have
been experimentally determined. Conversely, these characteristics and absolute transcription levels can be obtained
using relative RNA concentrations and RNA half-lives known from various experimental studies. In this case, the
“inverse problem” is solved with multi-objective optimization.
* Correspondence: [email protected]
Institute for Information Transmission Problems of the Russian Academy of
Sciences (Kharkevich Institute), 19 Bolshoy Karetny per, Moscow 127994,
Russia
© 2012 Lyubetsky et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
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Conclusions: In this study, we demonstrate that our model accurately reproduces all relevant experimental data
available for plant plastids, as well as the mitochondria of chordates. Using experimental data, the model is applied
to estimate binding intensities of phage-type RNA polymerases to their promoters as well as predicting terminator
characteristics, including polarization. In addition, one can predict characteristics of phage-type RNA polymerases
and the transcription process that are difficult to measure directly, e.g., the association between the promoter’s
nucleotide composition and the intensity of polymerase binding. To illustrate the application of our model in
functional predictions, we propose a possible mechanism for MELAS syndrome development in human involving a
decrease of Phe-tRNA, Val-tRNA and rRNA concentrations in the cell. In addition, we describe how changes in
methylation patterns of the mTERF binding site and three promoters in hypothyroid rat correlate with changes in
intensities of the mTERF binding and transcription initiations. Finally, we introduce an auxiliary model to describe
the interaction between polysomal mRNA and ribonucleases.
Reviewers: This article was reviewed by Dr. Anthony Almudevar, Prof. Marek Kimmel, and Dr. Georgy Karev
(nominated by Dr. Peter Olofsson).
Keywords: RNA polymerase interaction, RNA polymerase competition, Transcription, Circular DNA, mtDNA in
chordates, MELAS syndrome, Impact of DNA methylation, Hyposecretion of hormones, RNA interaction model,
Polysome and ribonuclease interaction model
Background
In this work, we use the model developed and applied to
study plant plastids [1] in the analysis of transcription in
the mitochondria of chordates. The model is generally
applicable to plastids and mitochondria, prokaryotes,
and nuclear DNA. We also propose an auxiliary model
describing a specific component of translation.
Many eukaryotes possess mitochondria, semiautonomous organelles with a highly reduced genome.
Animal mitochondria encode 22 tRNAs, 2 rRNAs and 13
proteins in a circular chromosome of 15–18 kbp. Transcription is conducted by phage-type RNA polymerases
homologous to the polymerases of phages T7 and T3.
Transcription initiation requires auxiliary protein factors
and occurs at up to five distinct promoters, producing
transcripts that can potentially be longer than the chromosome itself.
Among transcription factors are the mtTFA and
mtTFB proteins, which bind the polymerase at each promoter [2,3]. These factors detach from the polymerase
after transcription of 13 nucleotides. Several mtTFA isoforms are known that play a role in alternative splicing
[4]. Two mtTFB isoforms, mtTFB1 and mtTFB2, are
both involved in transcription initiation. However, the
role of these transcription factors is not considered in
this study.
Another essential initiation factor is mTERF [5], which
also mediates transcription termination in the form of
cooperative binding. This factor is an important component of our model. The properties of phage-type RNA
polymerases have previously been studied [6-9]. In particular, in a head-on collision, two RNA polymerases
approaching one another on the same DNA may pass by
one another [6]. We assume that in this case the
antisense mRNA forms a duplex and becomes inaccessible for the translation machinery.
Our study focuses on mitochondria of human
[Homo sapiens, Genbank:NC_012920.1], rat [Rattus
norvegicus, Genbank:NC_001665.2], and clawed frog
[Xenopus laevis, Genbank:NC_001573.1]. Data on the
mitochondrial genome of mouse [Mus musculus, Genbank: NC_005089.1] was also used as having the same
gene order. These model organisms were chosen due
to the availability of ample experimental evidence on
RNA concentrations and half-lives that can be used to
estimate gene transcription levels (transcription frequencies). The mitochondrial genomes of human, rat
and frog are given in Figure 1, Figure 2, Figure 3,
respectively. The RNA half-lives are described in
Additional file 1 (Section 1).
Structure and arrangement of mitochondrial promoters
Mitochondrial promoter locations differ substantially
among species. Table 1 shows experimentally determined locations of mitochondrial promoters in human,
rat, and frog. Human mitochondria possess three promoters, HSP1, HSP2, and LSP. Both the HSP1 and LSP
promoters contain the conserved box 5′-CANACC(G)
CC(A)AAAGAYA-3′ [10]. The transcription initiation
site is located inside the conserved box 6–8 bases before
3'-end. The transcription initiation site is located at position 561 in HSP1 (upstream from gene tRNA-Phe), at
position 646 in HSP2 (upstream from gene 12 S rRNA)
[11], and at position 407 in LSP [12]. The promoter
quality is affected by the regions: -16 – +7 for HSP1,
and −28 – +16 for LSP [13].
Rat mitochondria also possess three promoters [14].
The transcription initiation site is located at position
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Figure 1 Mitochondrial genome of Xenopus laevis. The entire circular DNA is presented sequentially as four rows. Genes in the heavy strand
(H-strand) are shown by green arrows, and genes in the light strand (L-strand) – by blue arrows. HSP1, HSP2 are two promoters in the H-strand.
LSP1, LSP2A, LSP2B are three promoters in the L-strand. mTERF symbolizes a binding site of the protein factor mTERF acting as transcription
terminator.
16298 in HSP1 (15 bases upstream from gene tRNAPhe), at position 66 in HSP2 (upstream from gene 12 S
rRNA), and at position 16193 in LSP.
Five promoters are known in frog mitochondria, HSP1,
HSP2, LSP1, LSP2A, and LSP2B, all lying upstream of the
tRNA-Phe gene [15,16]. Transcription is initiated at the
conserved box ACRTTATA. Relative transcription initiation intensities have been estimated for the wild type
and several mutant frog genotypes [17] (Table 2), and are
illustrated in our Table 2 for the reader’s convenience.
Figure 2 Mitochondrial genome of Homo sapiens. All symbols are the same as in Figure 1.
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Figure 3 Mitochondrial genome of Rattus norvegicus. All symbols are the same as in Figure 1.
Effect of protein factors and mtDNA methylation on gene
transcription levels
In early embryogenesis of frog a continuous increase of
mtTFA concentrations is observed [18] associated with
the elevation of gene transcription levels [19]. In the beginning of this period replication in mitochondria are almost arrested, and the initial pool of mitochondria is
distributed between the dividing cells.
Hormone concentration, patterns of mtDNA methylation [14,20] and mutations substantially affect gene
transcription.
Neither changes in mitochondrial gene expression
levels during development in human [21,22], nor
Table 1 Transcription initiation sites in mitochondria
Species
Sequence
Site
Position
Homo sapiens
[Genbank:NC_012920.1]
HSP1
561
Rattus norvegicus
Xenopus laevis
[Genbank:NC_001665.2]
[Genbank:NC_001573.1]
HSP2
646
LSP
(407)
mitochondrial gene expression levels in different tissues
of rat under different conditions [23,24] are currently
included in our model.
mTERF-dependent transcription termination
In human mitochondria, two terminators exist, each
possessing different termination mechanisms. In the first
mechanism, the mTERF protein binds to a 28 bp region
immediately downstream of the 16 S rRNA gene; and
located within the tRNA-Leu gene. This terminator is
considered to be polarized and blocks almost all RNA
polymerases on the DNA light strand but allows some
to move through onto the heavy strand. A negligible
portion of polymerases still passes through the terminator to transcribe the 16 S rRNA gene in the antisense
direction [19]. The second mechanism, the mTERFindependent transcription termination due to G-quadruplex, is described in Additional file 1 (Section 2).
HSP1
16298
HSP2
66
Table 2 Transcription initiation intensities relative to the
LSP1 promoter in frog mitochondria
LSP
(16193)
Promoter
HSP1
2102
HSP1
HSP2
2049
HSP2
60.0
LSP1
(2103)
LSP1
100.0
LSP2A
(2042)
LSP2A
16.6
LSP2B
(2033)
LSP2B
38.2
Positions in the complementary L-strand are shown in parentheses.
Data are from [17].
Intensity (%)
13.6
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In mammals, there are two hypotheses about mechanism of transcription regulation on the heavy strand
[5,14]. First, HSP1-initiated transcription may terminate
after the rRNA gene, and longer transcripts are initiated
at HSP2 only. Alternatively, all promoters may initiate
longer transcripts, and some polymerases stop at
mTERF regardless of the promoter.
Notably, in mammals mTERF binds cooperatively with
the termination site and the mTERF activator site in the
proximity of promoter HSP1, thus functioning as both
terminator and activator [11].
The termination site is conserved and located downstream of the rRNA genes in many animals [25]. Similarly mTERF homologs have been detected in the
nuclear genomes of many animals.
MELAS syndrome
The A ! G transition at position 3243, in the middle of
the mTERF binding site, decreases mTERF’s affinity for
DNA, thus causes a mitochondrial disorder. In human,
this mutation causes: (i) a small decrease of rRNA transcription (12 S and 16 S, at similar levels), (ii) a decrease
of tRNA-Leu concentration of up to 20%, (iii) a decrease
in tRNA-Lys concentration of up to 50%, (iv) a slight decrease in total mRNA, and (v) a noticeable change in the
amount of protein products [26]. These changes have to
be reproduced by any model.
Methods
Input data, main model and its parameters
Complete mitochondrial genomes of human, rat, and
frog analyzed in the study were obtained from GenBank,
NCBI, [27], Figure 1, Figure 2, Figure 3.
Here, the protein factor under consideration is the
multifunctional regulatory protein mTERF. G-quadruplex
implicates DNA regions, but cross-hairpins typical for
plastid [1] and bacterial genome DNA [28] are likely absent from the mitochondrial genomes of the animals
under study.
Let us recall our model of interaction between RNA
polymerases themselves, factors, and structures in DNA
or RNA [1]. RNA polymerases can potentially attempt to
bind with all promoters at a given locus. Each promoter
is characterized by real number λ, the intensity of
attempts to bind one of the surrounding polymerases.
Formally, λ is the parameter of a Poisson stochastic
process. An attempt is “successful” if the promoter is
not already occupied by a polymerase or any other factor, even partially. After binding, polymerases move
along the DNA strand and terminate at collisions with
oncoming polymerases. They can also terminate at
encountering protein factors or secondary structures.
The abort process is not considered, as only phage-type
RNA polymerases are present in mitochondria.
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Similarly, protein factors attempt to bind their target
sites. An attempt is “successful” if the site is not occupied by a polymerase or another protein. If a polymerase
encounters a site bound with a protein factor, it either
passes through and the protein-DNA complex dissociates, or it terminates and the complex survives. The
analogous situation for cross-hairpins [1], is beyond the
scope of the current study. Time intervals between any
successive events in the model were estimated from distributions of stochastic and deterministic (polymerase
movement) processes described above and were then
summed up. Thus, each event can be described with a
modeled real time commencing with the start of all
modeled processes. The modeled real time of course
does not coincide with the computation time, but it
allows the transcription level (transcription frequency)
for each gene to be computed in terms of the organism’s
time.
The model has four fixed parameters: (i) the elongation rate (which is constant among chordates), (ii) the
size of the phage-type RNA-polymerase, and (iii) the
ratio of polymerases terminating at a G-quadruplex, (iv)
the geometry of mtDNA, i.e., the location of genes, promoters and terminators. These parameters are discussed
in the next section.
Unknown parameters are the transcription initiation
(polymerase binding) attempt intensities at each promoter, mTERF protein terminator binding attempt intensities, and the conditional (provided that mTERF is
already bound) probabilities p and q of passing the
mTERF-dependent terminator in both directions. The
model contains no other parameters. All gene transcription levels are estimated in the model using the values of
these parameters.
The model software allows the user to specify physical
time of modeling along a trajectory, the model run-up
time, the number of trajectories to average, etc. The
main executing parameters are the number of processors
and time to halt modeling and create a checkpoint.
The solution to the model is a set of parameters, including intensities of binding attempts to all existing
promoters, conditional probabilities p and q of passing
the mTERF-dependent terminator in both directions,
and intensity λ of mTERF binding attempts. Here λ
incorporates the process of spontaneous dissociation of
the mTERFDNA complex. Table 3 shows conditional
probabilities p and q (passage probabilities) inferred
under two RNA polymerase movement rates.
The modeling procedure details are described in
Additional file 1 (Section 3).
Comparing the model and experiment requires knowledge of RNA half-lives. This was not considered in the
original model description [1]. Notably, for many genes,
the transcript concentration relative to the null time
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Table 3 The mTERF terminator passage probabilities on
the heavy and light strands
Rate, nt/s
p
q
L1n
(Frog1)
L1n
(Frog2)
L1n
(Frog3)
L1n
(total)
500
0.0164
0.0056
11.243
3.193
0.043
2.098
200
0.2165
0.0015
10.844
3.240
0.309
2.235
The minimum of L1n for three frog specimens, and minimum of L1n(total) are
shown. For comparison, the same minima are given for the RNA polymerase
elongation rate of 200 nt/s.
point or to a reference RNA concentration (relative
RNA concentration), and sometimes the RNA half-life is
known in a stationary state.
This allows a gene’s transcription level to be correlated
with its RNA concentration and half-life.
In frog, the RNA concentration uij is known for the jth gene at the i-th time point relative to its RNA con2Z t
z
centration at null time point, i.e., uij ¼ 2Zojij tjj ¼ zojij , where
zij is the transcription level of the j-th gene at the ith time point, tj is the RNA half-life of the j-th gene.
Absolute half-lives tj are unknown. Table 4 provides
experimental ratios uij (errors not estimated) and
z
ratios zojij of mean values zij, and they are compared
to each other. The values zij themselves are in
Additional file 2.
In human, RNA concentration uj relative to the reference gene ND1 and RNA half-lives are known in a sta2z t
tionary state, i.e., uj ¼ 2z0j tj0 , where zj is the transcription
level of the j-th gene, and tj is the RNA half-life of the jz
th gene. The ratio z0j is estimated in the model and compared with the experimentally known
t0
uj ;
tj
ð1Þ
in Table 5, upper rows. The special case of gene COX1
is discussed in Additional file 1 (Section 3).
In both rats, mRNA concentrations of the genes
COX1, ATP6/8, COX3, ND4, ND5, CYTB were measured individually relative to 16 S rRNA. Each ratio in
the hypothyroid rat was calculated as percentage of the
corresponding ratio in the euthyroid rat, Table 6. Thus,
ðzj h tj h Þðz0 e t0 e Þ
the experimentally measured ratio is uj ¼ ðz h t h Þ z e t e ,
0 0 ð j j Þ
where zj is the transcription level of the j-th gene in
hypothyroid (h) or euthyroid (e) rats, j = 1,. . .,6; z0 is the
16 S rRNA transcription level; t are half-lives specific to
j, e, h. Therefore, the value
zjh z0e
z0h zje
estimated in the model
was compared with the experimentally obtained
uj t0h tje
tjh t0e
ð2Þ
The numerator and denominator of the ratio are both
unknown.
Note that, when comparing gene transcription levels
in model and experiment, the choice of functional is not
straightforward. Consider the natural functional
X xji yji ;
ð3Þ
L1 n ¼
ji max xji ; yji
Where xji and yji are relative transcription levels in
gene j at time i (if no time data, the i index is omitted)
in the model and the experiment. To compare data on
three experimental frogs simultaneously, metric (3) can
be generalized:
!
3
X
1 X xji yji L1 nðtotalÞ ¼
ð4Þ
nk s ji max xji ; yji
k¼1
where nk is the “dimension” of compared data, s ¼
3
X
1
. The dimensions are, n1 = 54 (nine time points,
n
k¼1 k
six genes), n2 = 36 (six time points, six genes), n3 = 10
(five time points, two genes), respectively.
We also considered some other functionals, listed in
Additional file 1 (Section 4), which produced similar
solutions. Hereafter, only functionals (3)–(4) are
discussed.
Fixed parameters of the main model
All model parameters are listed in the previous section.
As for the mtDNA geometry, sequence data was
obtained from GenBank, NCBI, [27]. Multiple alignments were constructed with MEGA 5 [29], in order to
detect terminator sequences.
The ratio of polymerases terminating at a Gquadruplex is discussed in Additional file 1 (Section 1).
To be formal, we should include the RNA polymerase
elongation rate in unknown parameters, and the rate
should also be varied. Because of large computations, in
current study we examined only two values of the rate,
200 nt/s and 500 nt/s. The rate is unlikely to be less
than 200 nt/s, but values greater than 500 nt/s are possible (see the next paragraph). A preliminary prediction
of the model is that the polymerase elongation rate is
500–800 nt/s.
The elongation rate of phage type RNA polymerase (NEP)
The NEP rate is a key parameter in our model, but no
experimental data of this rate is available to the best of
our knowledge. The rate of replication fork progression
in E. coli is 1500 nt/s, which is assumed to be the maximal NEP rate. A lower bound for this rate can be
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Table 4 Agreement between the model and experiment for three frog specimens
time
mTERF LSP1
Frog1
ND1
COX2
mod exp dev
mod exp dev
mod exp dev mod exp
ATP6/8
ND4
1.0
1.0
1.0
1.0
Egg
0.0157 0.0034 1.0
1.0
+5 h
0.0448 0.0089 1.0
1.1
−12
0.9
0.8
+14
0.9
0.9
+1
+10 h
0.0872 0.0157 1.2
1.3
−5
1.1
1.1
+1
1.1
0.7
ND6
CYTB
dev mod exp dev mod exp dev
1.0
1.0
1.0
1.0
0.9
2.1
−59 2.4
2.4
+56 1.0
2.3
−57 4.1
4.0
1.0
1.0
−1
0.8
0.7
+19
+2
0.9
0.6
+50
+14 h
0.0793 0.0173 1.7
2.3
−26
1.6
1.6
−3
1.5
1.3
+18 1.4
3.0
−53 4.4
4.4
0
1.2
1.2
+3
+16 h
0.0960 0.0209 2.0
2.9
−31
1.7
1.4
+24
1.7
1.3
+31 1.5
4.3
−65 5.6
5.8
−4
1.3
1.3
+2
+18 h
0.0542 0.0157 2.1
3.2
−34
1.9
1.7
+14
1.9
1.9
+1
1.8
4.5
−60 4.4
4.2
+4
1.6
1.3
+25
+20 h
0.0655 0.0157 1.8
3.0
−41
1.6
1.4
+13
1.6
1.8
−12 1.5
4.6
−68 4.2
4.2
0
1.3
1.2
+8
−4
+49
7.4
6.5
+14 6.4
16.1
−60 12.9
12.2 +5
5.3
5.2
+2
26.0
26.1 0
23.8
60.3
−61 18.6
18.6 0
20.2
23.4 −14
48.3 −6
43.3
17.4 −4
38.8
+23 h
0.0721 0.0492 9.4
9.7
7.6
5.1
+48 h
0.0542 0.0872 29.3
26.6 +10
26.2
13.4 +96
+96 h
0.0407 0.0960 48.1
48.7 −1
45.3
time
mTERF LSP1
ND1
Frog2
mod exp dev
Egg
0.0089 0.0041 1.0
1.0
20.9 +117 45.4
COX2
ATP6/8
104.2 −58 16.7
ND4
mod exp dev
mod exp dev mod exp
1.0
1.0
1.0
1.0
ND6
39.3 −1
CYTB
dev mod exp dev mod exp dev
1.0
1.0
1.0
1.0
1.0
1.0
+6 h
0.0045 0.0023 1.2
1.3
−8
1.2
1.0
+22
1.2
1.3
−5
1.2
1.4
−12 0.7
0.7
+6
1.2
1.2
+3
+9 h
0.0073 0.0045 1.3
1.5
−14
1.3
1.3
−1
1.3
1.2
+8
1.3
1.6
−19 1.1
1.1
+1
1.3
1.3
−1
+20 h
0.0157 0.0157 3.8
4.6
−17
3.7
3.7
+1
3.7
3.7
+1
3.7
3.7
0
2.8
2.8
−2
3.6
4.0
−11
+30 h
0.0157 0.0230 7.2
7.2
0
7.1
6.8
+4
7.1
8.1
−13 7.0
6.2
+14 3.7
3.7
0
6.8
8.1
−17
+48 h
0.0407 0.1056 20.5
19.5 +5
19.7
19.7 0
17.7
+8
8.6
8.4
+2
17.3
23.1 −25
6.1
6.6
8.0
4.9
+36 2.4
2.3
+3
6.6
6.6
+7 days 0.0041 0.0073 6.5
time
mTERF LSP1
Frog3
+7
−18
16 S
ND6
mod exp dev
mod exp dev
Egg
0.0960 0.0026 1.0
1.0
1.0
1.0
+5 h
0.0407 0.0050 2.2
2.2
+0.9 2.2
2.2
0.0
+14 h
0.0230 0.0081 5.0
5.0
0.0
4.5
4.5
−0.2
+20 h
0.0038 0.0028 5.9
6.0
−1.3 4.0
4.0
+0.5
+28 h
0.0336 0.1056 92.2
92.0 +0.2 25.1
25.0 +0.4
+48 h
0.0143 0.0306 44.1
44.0 +0.2 15.0
15.0 +0.3
19.6
28.7 −32 19.1
6.6
8.5
−22 6.7
+1
In columns: “time” gives absolute times of embryonic development; “mTERF“ and “LSP1” are intensities of the mTERF factor binding with the termination site and
cooperative binding with the promoter, respectively, under the polymerase rate of 500 nt/s; further columns are modeled (mod) and experimental (exp) gene
transcription levels and their relative deviations (dev) obtained with equation (9). Variables are in bold.
estimated from the ratio of length E of the first exon
and length I of the first intron in protein-coding genes
in plastids of the Streptophyta lineage. In plastids, transcription and translation processes are concurrent, and
transcription of the first intron must end before translation of the first exon starts. Therefore, in the absence of
translational regulation, the ratio of the polymerase and
ribosome movement rates must exceed (E + I)/E. Such
regulation was predicted for genes atpF, clpP, petB, petD
[30]. For genes with short first exons, a delay in translation initiation must occur. Therefore, to estimate the
lower bound of NEP rate, only genes unlikely to possess
such regulation or trans-splicing should be used (e.g.
rpoC1, infA, ndhA, ndhB in Arabidopsis thaliana).
NEP-transcription is experimentally shown for rpoC1
[31] and can be suggested for other genes because of
the lack of quality PEP promoters. In different Streptophyta species the maximal ratios are 1.08 (infA in
Cucumis melo), 3.71 (ndhA in Chara vulgaris), 3.75
(rpoC1 in Zygnema circumcarinatum), 3.93 (ndhB in
Olea europaea). The greatest of these ratios, 3.93, corresponds to the lowest NEP rate of 177 nt/s. Higher
ratios are found in algae with uncharacterized transcriptional processes: 7.86 (rpl2 in Chara vulgaris), 7.94
(ycf3 in Zygnema circumcarinatum), 10.27 (ycf66 in Zygnema circumcarinatum). The greatest ratio suggests an
NEP rate of 462 nt/s. Further knowledge of the NEP
rate can improve the model accuracy.
Therefore, the lowest NEP rate of 500 nt/s and the
control rate of 200 nt/s were assumed. The model implementation enables the NEP rate and other parameters
to be varied.
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Table 5 Agreement between the model and experiment for healthy human and human with MELAS syndrome
Solution parameters for healthy human
Transcription levels (relative to ND1 gene) in the model (1st row) and experiment (2nd row)
LSP
HSP1
HSP2
mTERF
R
L1n
ND2
COX1
COX2
ATP6/8
ND3
ND5
CYTB
0.0031
0.0031
0.0126
0.6456
23.955
1.945
1.00
1.00
1.00
0.96
0.96
0.96
0.96
Experimental
estimates
Transcription level:
1.40
1.04
1.72
0.91
1.04
1.86
2.31
Error (if statistically independent):
±0.23 (1.8)
±0.52 (0.1)
±0.61 (1.2)
±0.43 (0.1)
±0.12 (0.7)
±0.56 (1.6)
±0.56 (2.4)
±0.40 (1.0)
±0.82 (0.1)
±0.95 (0.8)
±0.71 (0.1)
±0.20 (0.4)
±0.99 (0.9)
±1.01 (1.3)
Deviation from the experiment,%:
Error (if statistically dependent):
−29
−4
−42
+5
−4
−48
−58
Solution parameters for MELAS case
Changes of transcription levels in the model
0.0031
0.0004
0.0126
0.5336
24.333
Phe
12 S
Val
16 S
Leu
Lys
CYTB
3.84
1.20
1.20
1.20
1.16
1.22
1.17
Inferences are obtained under the polymerase rate of 500 nt/s and the values of p, q estimated for the frog.
Left panel: the solutions for healthy human (upper part) and human with MELAS syndrome (lower part). Variables and predicted values are in bold. Intensities decrease 7.75 times for HSP1 and 1.21 times for mTERF;
RNA/RNA− =1.18, R = 24. The L1n minima differ by 2.4% if general conditions are only imposed.
Right panel (upper part): Transcription levels of seven genes (relative to ND1 gene) in the model and experiment for healthy human. Experimental errors were estimated for both cases of statistically independent
measurements and counter assumption. In the latter case, the model results are within experimental error, except for the CYTB gene. Differences between model and experimental estimates are shown in parentheses
in the error radius units.
Right panel (lower part): For selected genes, decrease of their transcription levels in the model in MELAS case comparing to healthy human. The transcription level of tRNA-Phe drops 3.8-fold, of 12 S and 16 S – 1.2fold. The Leu and Lys transcription levels decrease 1.2-fold, which agree with experimental observations.
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Table 6 Experimental data on mitochondrial transcripts in rat
Gene
Euthyroid
Normalized mRNA/rRNA ratio
16 S
Hypothyroid
Half-life (min)
Normalized mRNA/rRNA ratio
44.48 ± 6.34
Half-life (min)
87.50 ± 27.52
COX1
100 ± 16
84.41 ± 27.49
86 ± 13
235.12 ± 48.68
ATP6/8
100 ± 19
78.14 ± 21.05
59 ± 9
277.52 ± 31.58
COX3
100 ± 19
78.14 ± 21.05
59 ± 9
277.52 ± 31.58
ND4/4 L
100 ± 16
84.41 ± 27.49
86 ± 13
235.12 ± 48.68
ND5
100 ± 25
46.00 ± 10.41
52 ± 11
60.52 ± 5.92
CYTB
100 ± 27
63.70 ± 7.82
57 ± 7
204.30 ± 28.64
Data are from [14]; the ± signs stand for standard deviations. For each gene, mRNA/rRNA ratios were normalized with euthyroid rat values taken for 100%.
The size of phage type RNA polymerase (NEP)
The NEP size is assumed equal to the size of the phage
T7 polymerase. The promoter size was experimentally
identified as −14 to +1 bp (relative to the transcription
initiation site) in mutagenic studies of the phage T7 NEP
ortholog in tobacco chloroplasts [7]. The −15 position
indicates a slight effect on the promoter quality [7].
Footprinting studies suggest that 15 DNA bases are occupied by NEP or that 11 bases are unpaired [8]. The estimate of 15 bases was obtained in X-ray structure
analysis of the phage T7 polymerase [9]. The current
study assumes the NEP occupancy to be −15 to +1.
Model of a polysome and ribonuclease interaction
(auxiliary model)
To explain the MELAS phenotype, an auxiliary model
was developed that describes the interaction between
polysomal mRNA and ribonuclease.
The following expression describes time τ of any
mRNA half-life, in terms of elementary probabilities (see
Additional file 3):
1
τ ¼ ð1 þ dλÞ expðwλÞ ln2;
μ
ð5Þ
νN
is the intensity of ribosome binding to
where λ ¼ 1þαN
its site, ν is a specific intensity under low N, where N is
the number of ribosomes in a healthy human mitochondrion, α is a Michaelis-Menten dependency parameter
(saturation over λ occurs at high N and equals αν ). Then,
w is the ratio of the linear size h (in codons) of the ribonuclease on mRNA to the rate V of the ribosome elongation (V = 15 codons per second, h = Vw = 15w), d is
the ratio of size h1 of the ribosome on mRNA to the
ribosome elongation rate V (h1 = 10 codons, h1 = Vd), μ
is the interaction intensity of the ribonuclease with a
specific mRNA site that leads to RNA cleavage. It is
clear that w cannot be less than 1/15 s, and it is likely
less than 4/3 s; N depends on the expression of other
genes, especially ribosome genes, its value is inferred in
the main model as absolute concentrations of 12 S or
16 S rRNA. Parameters ν, α and μ depend on the mRNA
sequence. Unfortunately, values of ν and α are unknown.
Note that ν and α depend on the ribosome binding site,
i.e., in the general case those are νj and αj over j RNAs.
It is plausible that μ is the same for healthy and diseased
humans.
Although here only the ribonuclease is considered as a
factor in mRNA cleavage, other factors can be considered analogously.
Similarly, in diseased human, the mRNA half-life τ′ is
expressed in terms of N′, the number of ribosomes in
the mitochondrion, thus
h i τ
0
1 þ dλ
¼ 0:
0 exp w λ λ
τ
1 þ dλ
ð6Þ
The left hand of (6) is evidently more than one, τ/τ′ >
1. For at least one mRNA, the experimentally expected
interval of τ/τ′ is 1.5–3 [26]. For that mRNA, it follows
from equation (6), that a small change in the absolute
number N of ribosomes greatly influences the mRNA
half-life, and thus the amount of the corresponding protein. The model solution confirms that the half-lives of
rRNA and most mRNAs are similar in healthy and diseased humans. The disease phenotype might require a
sharp decrease in half-life of a single (even short) mRNA
(see item 6 in Additional file 1, Section 3). This suggests
a plausible explanation for the MELAS phenotype.
The intensity of RNA decay by the ribonuclease is
given by:
κ¼
μ
expðλwÞ:
1 þ dλ
ð7Þ
Each mRNA was assumed above to possess a single
ribonuclease site. To the best of our knowledge the actual number and arrangement of cleavage sites on RNA
remain to be determined. However, if the average number of cleavage sites per RNA for a given gene among all
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Page 10 of 18
mitochondria in a tissue is k, then in the above equation
κ is substituted by κk. This conclusion and equations (5)
to (7) are derived in Additional file 3.
An RNA window is defined as a region of a minimum
length h = Vw, that separates two neighboring ribosomes, where Vw is the linear size of the ribonuclease
on mRNA.
By combining equations (5) and (6) for different
RNAs, νj can be expressed in terms of w.
Comparison to the experiment
The distributions of variables uj ; t0 ; tj are not experimentally known. This precludes us from estimating confidence for experimental values (1) or (2) on the basis of
probability-theoretical methods which are usually applied to compare predictions with experiments. However, absolute errors can be used instead. Let Δ be an
absolute error of b (expressed as (1) or (2)) and a is a
model value for the same expression, then it can be verified if:
}a belongs within b Δ}
ð8Þ
The error Δ of expression value (1) or (2) is then estimated trivially [32] by using one of two common empirical relations. First, the error of an algebraic sum is
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Δðx yÞ ¼ ΔðxÞ2 þΔðyÞ2 if summand errors are statistically independent. Alternatively, Δðx yÞ≤ΔðxÞ þ ΔðyÞ.
The error of product x y or ratio x/y is Δðx∘yÞ ¼ ðx∘yÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðΔðxÞ=xÞ2 þ ðΔðyÞ=yÞ2 if member errors are statistically independent. Otherwise, Δðx∘yÞ≤ðx∘yÞ ðΔðxÞ=xþ
ΔðyÞ=yÞ. Therefore, either a hypothesis of statistical independence is assumed to be true and equalities apply,
or inequalities are used, in which case expression (8)
becomes uncertain. Fortunately, both cases produce
similar results.
The experimental and model values of (1) and (2) and
their errors are given in Table 5 and Table 7, assuming
that constituent errors are either statistically independent, or not. The model predictions fall within the error
intervals b 1:3Δ and b 2:4Δ , respectively (Table 5
and Table 7).
Importantly, experimental and predicted solutions
from Table 4, Table 5, Table 7 differ “insignificantly” in
another respect. The percentage concordance between
values a and b (empirical and predicted) can be
expressed as:
ða bÞ 100=b:
ð9Þ
The sign of this value indicates a decrease or increase
of a relative to b. As a rule-of-thumb, changes in gene
transcription level that are between −50% (i.e. halving)
and +100% (i.e. doubling) are considered “insignificant”
[33]. By this measure, almost all model predictions differ
insignificantly from experimental data (Table 4, Table 5,
Table 7).
Model implementation
The model was implemented in C++ in two versions
(command line interface and GUI) and is distributed
freely under the GNU General Public License v.3 on the
web page [34]. Similar to the earlier release [1], it is an
event-driven automaton that simulates large combinations of interacting stochastic and deterministic processes in a DNA locus against modeled physical time.
RNA polymerase binding is modeled as a stochastic
process. The subsequent polymerase elongation in this
study is modeled as a deterministic process with constant rate. The following collision events are modeled: (i)
a polymerase or factor attempts to bind a previously occupied site, (ii) secondary structures attempt to form
within a bound site, and (iii) two oncoming polymerases
attempt to process the same nucleotide. Scenarios and
rules of the collision’s resolution are customizable model
options.
The events in the model are handled chronologically,
and ordered in a complex system of partially ordered
queues. The program performance depends heavily on
the rate of serving the queues.
Polymerase interaction was previously studied within
short loci of a few thousand base pairs [1]. Here we
model transcription along complete mitochondrial genomes of up to 18 kbp in length. The circular arrangement of this DNA introduces a novel scenario:
polymerases can continue transcription until they
complete several circles and collide, which considerably
increases the number of simultaneously modeled events.
Another important aspect of our model is that we
considered phage-type RNA polymerase. Although its
elongation rate is not experimentally known it is likely
higher than that of the bacterial-type polymerase. In
plastids, transcription is carried out by polymerases of
both types. However, a faster polymerase does not make
a difference, as it cannot outpace or influence the slower
polymerase. In this study, elongation rates of 200 nt/s
and greater were assumed, which is an order of magnitude higher than in the earlier modeling attempts. This
led to a higher rate of access to larger event queues and
therefore to reduced performance. Queue processing
was considerably improved in the current implementation by changing from a linear event queue to a system
of partially ordered queues, which allowed us to attain
previously observed levels of performance [1].
In our previous implementation, any polymerase collision with a factor or secondary structure terminated
transcription [1]. The current model implements a new
class of objects, protein terminators with nonzero
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Table 7 Agreement between the model and experiment for euthyroid and hypothyroid rats
LSP
HSP
mTERF
R
L1n
Transcription levels in hypothyroid rat relative to euthyroid one in the model (upper) and experiment (lower)
COX1
ATP6/8
COX3
ND4
ND5
CYTB
0.1056
0.0721
0.9453
30.605
1.736
0.666
0.641
0.646
0.622
0.614
0.613
0.1056
0.0336
0.9453
30.637
0.61
0.33
0.33
0.61
0.78
0.35
Error
(if statistically independent):
±0.35 (0.2)
±0.17 (1.9)
±0.17 (1.9)
±0.34 (0.0)
±0.42 (0.6)
±0.17 (1.5)
Error
(if statistically dependent):
±0.79 (0.1)
±0.39 (0.8)
±0.39 (0.8)
±0.79 (0.0)
±0.97 (0.2)
±0.39 (0.7)
Deviation from the
experiment,%:
+9
+94
+96
+2
−21
+75
Inferences are obtained under the polymerase elongation rate of 500 nt/s and the values of p, q estimated for the frog.
Upper left part: the solutions for the euthyroid (upper row) and hypothyroid (lower row) rats. Variables and predicted values are in bold. Note that HSP =
HSP1 + HSP2.
The right part: experimental data and model predictions. Experimental errors were estimated for both cases of statistically independent measurements and counter assumption. In the latter case, the model results are
within experimental error. Differences between model and experimental estimates are shown in parentheses in the error radius units.
Page 11 of 18
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passage probabilities in both directions; these probabilities p and q are the terminator parameters mentioned
under Methods.
The model constructs an individual trajectory in the
event space and estimates gene transcription levels along
the trajectory. Under the same set of model parameters,
the levels are averaged over multiple trajectories. Computations can be effectively parallelized on a cluster with
MPI v. 1.2 or newer. In this study the results were
obtained on 2048 CPUs at the MVS-100 K supercomputer
of the Joint Supercomputer Center of the RAS [35].
The inverse problem was solved by multi-objective
optimization. For example, the response landscape for
functional (4) is complex, with numerous valleys and
local minima, which excludes the use of standard (e.g.
gradient-based) local minimization techniques and
requires heuristic solutions. Our approach is based on
the following effective strategy: promoters in both
strands of mtDNA are concentrated in a region containing only Phe-tRNA genes in human and rat, which produces opposite flows of polymerases that compete
mostly outside this region. Therefore, a lack of transcription on one strand indicates that it is blocked by a heavy
flow of polymerases along the opposite strand. Under
the general assumption of non-zero transcription level,
further increase of promoter binding intensities on the
successful strand is not reasonable and does not lead to
a better solution. This fact considerably reduces computational complexity. For any current set of model parameters, promoter binding intensities are varied in each
direction only until transcription stops for one of the
genes. This optimization strategy is referred to as “active
search”.
Results
Given an RNA polymerase transcription rate of 500 nt/s,
one solution was obtained for each model organism, the
Page 12 of 18
frog (three embryos), human (healthy and diseased), and
rat (eu- and hypothyroid).
In all cases the mTERF terminator passage probabilities (i.e., the portion of RNA polymerase passages
through bound mTERF) were p = 0.0164 on the heavy
strand and q = 0.0056 on the light strand, indicating a
triple-fold polarization of the terminator.
In frogs the LSP1 promoter binding intensities mostly
increase with time (Table 4 and Figure 4). Predicted and
experimental transcription levels are in good agreement,
with the exception of the estimate obtained with equation (9) for the first frog. In this case, at 96 hours of development the difference slightly exceeds +100%; for
gene ND4 at all time points the difference slightly
exceeds −50%, Table 4.
In healthy human the LSP, HSP1, HSP2 and mTERF
binding intensities are 0.0031, 0.0031, 0.0126, and
0.6456, respectively (Table 5). In the case of MELAS
syndrome the HSP1 and mTERF intensities drop
7.75- and 1.21-fold and become 0.0004 and 0.5336, respectively. The ratio R of gene 12 S to gene COX2 transcription levels is 24, and the ratio RNA/RNA− of
weighted total RNA concentrations in healthy and diseased human is 1.18 (these denotations, R and RNA/
RNA–, are explained and justified in Additional file 1,
Section 3). Transcription levels of tRNA-Phe and 16 S
rRNA dropped in diseased human 3.84- and 1.2-fold,
respectively. The decrease in tRNA-Leu and tRNA-Lys
transcription was 1.2-fold, which is within experimental
error. The functional value for optimal solution under
all imposed additional conditions vs. only general conditions (ref. definitions in Additional file 1, Section 3) differs
by 2.4%.
All predicted and experimental transcription levels in
healthy human are within experimental error, except for
CYTB, for which this error is exceeded by 29%. In
addition CYTB exhibits a difference of around −50% in
Figure 4 LSP1 promoter binding intensities plotted against time for three frog embryos. The LSP1 promoter binding intensities mostly
increase in agreement with [19].
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Figure 5 Transcription numbers predicted during 9 hours of modeled physical time. A, B, C – frog embryos 1, 2, 3; D – healthy human;
E – human with the MELAS phenotype; F – euthyroid rat; G – hypothyroid rat. Genes on the heavy strand are in green, on the light strand – in
blue. Shaded are mean values, not shaded – standard errors.
healthy human as estimated with equation (9), Table 5.
We will revert to the case of CYTB in the Discussion.
In rats, we defined HSP as the total intensity of binding attempts to promoters HSP1 and HSP2, i.e. HSP =
HSP1 + HSP2. In euthyroid rat, the binding intensities
for LSP, HSP and mTERF are 0.1056, 0.0721 and 0.9453,
respectively. In hypothyroid rat, HSP drops to 0.0336
(Table 7). The ratio R of 12 S and COX2 transcription
levels is 30.605 in euthyroid rat and slightly increases to
30.637 in hypothyroid one. Similarily, differences in predicted and experimental transcription levels between the
eu- and hypothyroid animals are within experimental
error. Differences obtained with equation (9) are insignificant, Table 7.
The model’s solutions and comparisons to experimental values are presented in Table 3, Table 4, Table 5,
Table 7. The predicted absolute transcription levels of
all genes are illustrated in Figure 5 and detailed in
Additional file 2. Importantly, most predictions fall
within experimental error. This definition, however,
needs further justification. Specifically, a more accurate
error estimation would require knowledge of the distribution of the experimental measurements (see Section 4
of the Methods).
Discussion
The high degree of evolutionary conservation of the
mTERF factor and its binding site [25] allows estimates
of parameters p and q (the terminator passage probabilities in both directions) to be extrapolated among chordates. However, other parameters should be interpreted
with caution. For example, the mouse possesses the terminator D-TERM, which is unknown in human and rat,
between promoters LSP and HSP1 in the 5'-leader region of the tRNA-Phe gene [36]. Homologs of this terminator cannot be detected by sequence similarity even
across close relatives (see Additional file 1, Section 5).
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Our model’s predictions agree well with experimentally
derived transcription levels in the mitochondria of frog,
human and rat, including disease states such as the
MELAS syndrome in human and the hypothyroid condition in rat.
The model predicts the intensities of polymerasepromoter binding, properties of the mTERF transcription
terminator and absolute transcription levels of all mitochondrial genes (Figure 5 and Additional file 2). This is
in contrast to the experiment which only provides relative measures of transcription of selected genes. An argument in favor of our model is the fact that all terminators
are important in the prediction of transcription. Notably,
excluding any of the terminators from modeling leads to
predictions that intuitively are inadequate, e.g., transcription of only one DNA strand.
With a polymerase transcription rate of 500 nt/s, the
model conforms better to the experiment (i.e. most predictions fall within experimental error) than with a rate
of 200 nt/s. However, the rate choice needs further justification both in silico and in the experiment.
The concentration of the mtTFA transcription factor
increases monotonically during early embryogenesis in
frog [18]. mtTFA is a universal activator, and the
expected increase of its binding intensities to all promoters is accurately predicted by the model (column LSP1,
Table 4). There is no similar evidence for the mTERF
factor, and the model does not predict the monotonic
change either.
In mammals, the efficiency of the HSP1 promoter is
predicted to be markedly greater than that of the HSP2
promoter on the heavy strand [5]. Notably, this result is
not supported in our model, where HSP2 is 4 times
more efficient than HSP1.
In human, the predicted absolute transcription levels
of protein-coding genes are unexpectedly low. One transcript is produced every 15–26 minutes, depending on
the gene (Figure 5, Additional file 2), while the entire
mitochondrial genome is transcribed in 33 s under the
polymerase elongation rate of 500 nt/s. Such selective
transcription agrees with previous measurements of absolute mRNA concentrations [37].
Furthermore, in human, transcription levels of proteincoding genes on the light and heavy strand are predicted
to be very similar. This suggests that polymerases that
initiate at promoters HSP1 and HSP2, and that pass the
mTERF terminator collide very rarely. Such counterflows of polymerases are also rare for LSP-initiated polymerases that passed the first G-quadruplex terminator.
There is negligible competition between free and bound
polymerases for the promoter, because under high
elongation rates and low binding intensities each promoter becomes available long before the next binding attempt. The probability that a promoter bound by a
Page 14 of 18
polymerase continues into the next circle is also low: in
9 hours of modeled time there are only 1 ± 1 such polymerases on the light strand and 23 ± 6 – on the heavy
strand (average values ± unbiased standard deviation
over n = 1000 trajectories). A possible biological explanation of such low polymerase competition in mitochondria of human is minimization of DNA damage during
collisions. In plastids, the bacterial-type polymerases are
considerably slower but collide frequently [1]. The
bacterial-type polymerases inherited by mitochondria
from their α-proteobacterial ancestors might have been
lost and replaced by faster phage polymerases to reduce
DNA mutation rate [38] (Part 3, Section 9.7).
In frog and rat, on the other hand, polymerase competition is more pronounced (see Figure 5 and Additional
file 2), which can be explained by differences in RNA
half-lives.
In human with MELAS syndrome, the model predicts
a 1.21-fold decrease of the mTERFDNA binding intensity and a 7.75-fold decrease of the HSP1 promoter efficiency. Transcription levels of tRNA-Phe and rRNA drop
3.84- and 1.2-fold, respectively, with possible implications for the MELAS phenotype. For example, consider
the influence of rRNA. Highly expressed mRNAs are
normally entirely covered by ribosomes that protect
them from adverse modifications. The decreased expression of rRNA predicted by our model may lead to
decreased ribosomal coverage of mRNAs and cause
increased transcript damage (see Section 3 of the Methods). Similarly, we can consider the influence of tRNAPhe: underexpression of tRNA-Phe attenuates translation
and, at the same time, widens the window between ribosomes on RNA (see Section 3 of the Methods).
Increased transcription of CYTB relative to the upstream genes reported from the experiment cannot be
predicted in the model. However, in thiamphenicoltreated cells the CYTB mRNA half-life shows a high degree of variation [39] (Table 2). This suggests a possible
experimental bias in measuring the time and thus the
CYTB transcription level. A slight discrepancy of 6% between the model and experiment for expression of the
human ND2 gene might be explained by a similar bias
in estimating half-lives [39] (Table 2).
In the model, intensities of the mTERF binding and
LSP transcription initiation are equal in eu- and
hypothyroid rats. In experiment, methylation of the
mTERF binding site remains unaffected, and of the LSP
promoter exhibits minimal changes [14], which agrees
well with the stability of their predicted binding
intensities.
Conversely, in the model, the total intensity HSP of
transcription initiation from promoters HSP1 and HSP2
is 2.15-fold lower in the hypothyroid rat. In experiment,
methylation of the HSP1 region changes considerably,
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but methylation of the HSP2 changes negligibly [14],
which also agrees with the predicted decrease of transcription initiation from promoters HSP1 and HSP2.
Conclusions
Our previous analyses [1] and current study demonstrate
that the model produces results in good agreement with
experimental evidence from plastids and mitochondria.
It accurately predicts the RNA polymerase binding intensities, transcription terminator characteristics, and
absolute transcription levels of all mitochondrial genes.
Individual gene transcription intervals are predicted to
be long in human (15–26 min) and rat (2–10 min), but
short in frog (8–25 sec). RNA polymerase competition is
shown to be negligible in the mitochondria of human
but evident in rat and frog, albeit much less intense
compared to that in plastids. Advantages of the phagetype vs. bacterial-type RNA polymerases in mitochondria
are suggested and discussed.
In hypothyroid rat, we describe how changes in
methylation patterns of the mTERF binding site and
three promoters correlate with intensities of the mTERF
binding and transcription initiations.
We have also proposed a polysome-ribonuclease interaction model (see Section 3 of the Methods) and factors
explaining the MELAS phenotype development in
human.
Additional files
Additional file 1: Supplement 1. RNA half-lives, mTERF-independent
transcription termination, modeling procedure.
Additional file 2: Supplement 2. Transcriptions predicted during 9
hours of modeled physical time.
Additional file 3: Supplement 3. Details of the auxiliary model.
Competing interests
The authors declare no competing interests.
Authors’ contributions
VAL, OAZ and AVS proposed the model, estimated its parameters, chose
source data and participated in the analysis of results. VAL, SAP and AVS
wrote Additional file 3 including derivation of formulas (5)–(7). LIR actualized
the model implementation: wrote software, performed the computations
and participated in the data preparation and analysis. OAZ performed the
sequence alignment, the statistical analysis and participated in the data
preparation and the computations. VAL, OAZ, AVS and LIR wrote the
manuscript. All authors read and approved the final manuscript.
Reviewers’ comments
Reviewer’s report 1 – Dr. Anthony Almudevar
The authors apply a parametric model for RNA polymerase interaction
during transcription, developed in an earlier paper [1], which successfully
predicted changes in gene transcription levels attributable to various
experimental perturbations. The model is applied to new data, and appears
to represent a significant extension of the model.
The reported predictions are quite interesting, but the paper as a whole
seems very hastily written, often consisting of a sequence of brief
paragraphs. This is especially true of Section [4. Modeling Procedure] on
Page 13. The paper needs to be significantly reorganized.
Page 15 of 18
Response. A sequence of brief paragraphs is justified by the intention to
concisely list various experimental data and to state what the model is
based on. Generally, the manuscript was reorganized (refer to response #2 to
Reviewer 2), and currently this sequence is distributed between the main
text and Supplement 1.
I think it would also be important to introduce a careful definition of the
model, introducing notation for all parameters and observations at one
place. The paragraph on page 8 beginning "The solution is a set of
parameters …" introduces λ, p and q, but more model elements are
introduced at various subsequent places in the paper (including the
Supplementary). It would better to see the full extent of the model,
including a clear definition of the parameter space, at this point.
Response. The model definition in Section 1 of the Methods is extended
and all model parameters are now grouped. Additionally, the software setup
and execution parameters are briefly described.
Quality of written English: Needs some language corrections before being
published.
Response. Done (see the last response to Reviewer 2).
Reviewer’s report 2 – Prof. Marek Kimmel
The model is an application of the previously published more general model
of interaction of RNA polymerases, protein factors and secondary structure
during transcription. Current version is limited to phage-type polymerases
and transcription factors in the mitochondrial genomes of human, rat and
frog.
Response #1. The current version of the model is more sophisticated: it now
accounts for a protein-dependent terminator (mTERF) and completion of
more than one circle on the mtDNA strand by RNA polymerases.
Furthermore, the polymerase interaction described in [1] was not necessarily
applicable to mitochondria, as the model behaves differently: it predicts high
polymerase competition in plastids, and negligible competition in human
mitochondria. Moreover, modeling the same mechanism in mitochondria
provides explanations for the MELAS phenotype and the phenotype caused
by hyposecretion of thyroid hormone. Although, different functional
phenomena were explained in [1].
The model is based on comprehensive data and a detailed analysis of
inferred parameter values. This is an interesting paper and it may be
ultimately published. However, as it is now, it is overloaded with detail on
one side and is missing some important information on the other.
Accordingly, in my opinion, most of the details included in Sections 4-6 of
the Background, should be moved to an online Supplement. Same is true of
most of Section 4 of the Methods. What is to remain in the main body, are
pointers to respective sections of the Supplement. Of Figures 1, 2 and 3, one
might remain in the paper body. Background section is missing a description
of the principles of the model. Instead, the authors seem to have decided
that it will be enough to cite an earlier paper and defer the description (in a
rudimentary form) to Section 5 of the Methods. This is unfortunate and I
think should be corrected by expanding this material to a full-fledged
description and moving it to the Background or to the beginning of the
Methods. The part of Discussion in pp. 23-24 belongs more to the model
description or to the Methods. After this "cleanup", the manuscript will be
much more legible.
Response #2. Sections 4, 6 of the Background and section 4 of the Methods
have been moved to Supplement 1 (Sections 1–3). Discussion in pp. 23-24
has been moved into the model description (Section 1 of the Methods). The
model description in Section 1 of the Methods has now been extended and
all model parameters grouped together.
Detailed remarks
Page 8. "Ann attempt is successful if the promoter is not occupied by a
polymerase or factor." This seems unclear; the factors are supposed to attract
polymerases (?)
Response. Transcription factors (proteins) binding DNA close to a promoter
often act as activators or repressors of transcription. Such factors are
common in plastids and mitochondria. In the mitochondria of metazoa and
some protozoa, the mTERF factor has an important function as both a
terminator (not bound to the promoter) and an activator. Modeling predicts
the intensities of mTERF binding and transcription initiation attempts for all
promoters that unlikely can be measured directly in the experiment.
Page 8, bottom. Define "contents".
Response. The "relative RNA content" is replaced with «relative RNA
concentration»: uij is the RNA concentration of the j-th gene at i-th time
point relative to the same concentration at the null time point, i.e., the ratio
Lyubetsky et al. Biology Direct 2012, 7:26
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of two concentrations sampled during an experiment. Notably, the
experiments do not allow measuring numerator and denominator of the
ratio.
Page 9. What are z0j and z0?
Response. Here zij is the transcription level of the j-th gene at the i-th time
point, z0j – transcription level of the same gene at 0-th time point, i.e., uij =
zij/z0j. Similarly, zj is the transcription level of the j-th gene, z0 – the
transcription level of the 0-th (reference) gene; we calculated zj/z0 to
compare it with the experimentally determined uj.
Page 9. "hypothyroid or euthyroid". Rat ?
Response. Indeed, rat. Corrected.
Page 9. The "lower and upper parts of the ratio" are usually called numerator
and denominator.
Response. Corrected.
Page 10. Define x and y.
Response. These variables x̄, ȳ have been eliminated.
Page 10. The list of "considered functionals" is superfluous.
Response. The list has now been moved to Supplement 1 (Section 4).
Page 11. The error analysis on this page is not very professional. A statistician
should be consulted at least regarding notation and references.
Response. Following the reviewer's suggestion, we improved Section 4 of
the Methods. According to Chapter 4, Section 4.3, of [28], “under the normal
distribution of measurements and a large sample size approximately 70% of
them belong within x̄ ± σ (x), where x̄ is the mean. Apparently, the standard
deviation σ(x) bears the previously discussed meaning of “error”. If
measurements of x are scarce, then their error is to be Δx that equals σ(x)”.
From [28], equality Δx = σ(x) is postulated also in cases of indefinite
distributions of measurements. Rules of estimating the error of the sum,
difference, product and ratio are given in Chapters 1, 2, 3 of [28], and are
stated in Section 4 of the Methods. Tables 8-9 now contain errors estimated
accordingly.
Page 14, top sentence. Is this stabilization a hypothesis, empirical
observation, or conclusion bad on modeling?
Response. It is the result of the modeling.
Page 31. Should more details be included in the legend of Figure 4?
Response. Corrected.
Table 10. This Table has important information, but probably will be better
off reworked into a bar chart. The table itself may be moved to the
Supplement.
Response. Table 10 has now been moved to Supplement 3 and replaced by
Fig. 5 in the main text.
Quality of written English: Not suitable for publication unless extensively
edited
Response. The authors recruited native speakers to edit the language
(mentioned in the Acknowledgements).
Reviewer’s report 3 – Dr. Georgy Karev (nominated by Dr. Peter Olofsson)
This work continues a recent paper of the same author's team [1], where a
mathematical model of interaction and competition of many RNA
polymerases transcribing simultaneously a DNA locus, was formulated and
developed. The model was based on the concept of interactions suggested
and developed by the authors. The concept has underlying "mechanistic
image": similarly to "hard spheres", RNA polymerases are connected with
DNA molecule and then move and come into collision with each other'
according to certain stochastic rules. Surprisingly, a computer model based
on this simplified mechanistic, not biochemical, description of interaction
processes with more or less universal parameters was able to predict the
expression levels of genes in different loci of plastids of plants in [1] and
mitochondria of chordates (frogs, rats and human) in the present paper.
Some other interesting phenotypic phenomena studied in both papers can
also be explained within the frameworks of the developed model.
Additionally, an interesting model of interaction between polysomes and
ribonucleases was formulated in the present paper.
The algorithmic realization used a representation of the model as a multiagent system; interactions between the agents were modeled using the
Monte-Carlo method. The behaviors of agents at every instant were
modeled by certain deterministic or stochastic processes, and corresponding
events happened randomly and asynchronously. The model uses nonuniform discrete time moments, which correspond to the moments of event
occurrences.
Each run of the program models a single realization of inter-cell processes.
Biological experiments use tissues composed from many millions of cells
Page 16 of 18
implying a result naturally averaged over all cells. Similar result form in silico
experiments can be obtained only by multiple run of the program and
subsequent averaging of the results. It demands a powerful computer
system and a long time for computations.
Overall, the work presents a complex model based on non-trivial biological
background and mathematical theory of stochastic processes, which was
realized as a program package for multiprocessor computer; the authors
used a cluster with 2048 processors and nevertheless the computations took
a long time. The model has explanatory properties and also allows
predicting the transcription levels of future experiments. The program
package, the manual and tutorial examples are available on the laboratory
site. It would be very helpful for readers if the manual would be preceded
by non-formal but complete enough description of the model.
Suggested changes:
The authors refer to their previous paper [1] for the model description, so it
would be helpful to give a summary of the model (background, main
assumptions, variables and processes).
Response. See response #2 to reviewer 2.
A logical path from biological motivation and "verbal" model to computer
model is not entirely clear from the text. In particular, mathematical scheme
of the model and especially the computer algorithm contain some
additional assumptions about interactions between deterministic and
stochastic processes that are accounted for the model; these assumptions
were realized in the program but they do not follow immediately from the
biological background. Supplementary materials contains enough
explanations for general understanding of the model but it would be helpful
to extend it or to give additional explanations for missing mathematical
details.
Response. Each point in the description of both models (Sections 1 and 3 of
the Methods) contains assumptions, including the Poisson nature of the
flows, discrete time, averaging over trajectories with the Monte-Carlo
approach, ordering the queue, modeling parameter settings (such as the 9
hrs time period to count transcriptions), etc.
A non-trivial proof of the auxiliary model (Supplement 2) is contained in
equation (*): the probability of RNA cleavage inside the spliceosome during
short time Δt in a window equals the product of μΔt + o(Δt) (the probability
exp(−λw)
of RNA cleavage in the window) and
(the probability of existence
1+dλ
of this window during Δt). The difficulty here is the formal assumption of a
stationary process (lack of cleavage) to describe the probability of opening
the window for a generally non-stationary process. A rigorous mathematic
study of this non-stationary process is sophisticated and is not discussed in
this work.
The model is a complex and rather specific system of interconnected
stochastic processes. Computer realization of complex systems and
computations even of a large number of individual trajectories cannot
guarantee that some important qualitative peculiarities of the system
behavior were not missed. Are any general mathematical results about
qualitative behavior of the model known by the authors?
Response. This is an important and interesting point. Unfortunately, many
particularly mathematical questions remain unanswered in this model, e.g.,
an explicit description of the stochastic process of effective (in terms of the
“intensity”) polymerase-promoter binding attempts even under a fixed
arrangement of only two opposite promoters.
A large number of deterministic or stochastic events that happen in random
time moments were taken into consideration in the model. Computer
realization of any flow of events demands some kind of ordering of the
events, because they are considered "one by one" in a computer. The way of
ordering of the event flow generally imposes some additional restrictions
and conditions in the initial model. The authors give only a few explanations
of how the ordering of event flow is organized in the computer model.
Response. Despite the fact that all processes in the model initiate at the
same “zero” time, the first polymerase binds to each promoter after a
random time determined by the Poisson process associated with that
promoter. It is unlikely that these precisely calculated times coincide for
different promoters, therefore subsequent discrete “moves” of all
polymerases also occur at different time-points, because they have the same
rate. Thus we may assume simultaneous events do not occur; in particular,
any collision always involves one moving object and one stationary object.
This is the sole assumption and it is physically plausible. This assumption
allows us to use non-uniform discrete time instead of continuous time. Once
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a future event is determined to occur at some time, it is inserted in the
priority queue according to that time. Each step of the modeled time
involves extraction of the next event from the queue top. The priority queue
must be capable of containing tens of thousands of time-ordered events
which causes serious algorithmic and computational difficulties. The
program performance depends heavily on the speed of the queue servicing.
Currently the queue is implemented as a binary heap, and the average
performance we reached in the model is one to two orders of magnitude
faster than modeled physical time.
The authors describe in good details the biological background of the
transcription process, which the model was based on. It would be helpful to
point out those phenomena and processes, which were not accounted in
the model. It would help to understand better the boundaries of
applicability of the model and possible ways to its improvement.
Response. A number of transcription-related processes are not considered in
this study. For example, the rate of the phage type polymerase is likely to be
affected by RNA primary and secondary structures. Transcription initiation
factors mentioned in the Background are not considered here. Termination
at a G-quadruplex is not modeled. Only the experimental ratio of
terminating polymerases is considered. The formation of a supercoil and ZDNA in front of the moving polymerase can lead to its remote interaction
with other protein factors.
The authors pointed out in their previous paper that nucleotide composition
of DNA was not accounted for in the existing version of the program (more
exactly, it was used only for other objects to be related with certain position
of the DNA sequence), and that the nucleotide composition of DNA can be
taken into account in the next versions of the program. By now having more
experience with the program, do the authors believe that more explicit
account of nucleotide composition is necessary or desirable, and if so, what
are the corresponding problems?
Response. The authors developed a model and its computer realization that
describe the effect of the RNA primary and secondary structures on the rate
and nature (pause, arrest, termination) of the polymerase movement [40].
However, incorporating these would lead to reduced performance of the
software in its current implementation.
The authors have concluded that the half-life of mRNA in human
mitochondria can be changed significantly even after small variations of
transcription intensity; for this reason it is not clear why they take equal halflife values for mitochondria of a healthy human and a human with MELAS
syndrome.
Response. This prediction is qualitative. At least, both our models predict
slight changes in transcription of selected genes associated with the MELAS
phenotype.
Two models are described in this study: the main model of RNA-polymerase
interaction, and the auxiliary model of mRNA-ribonuclease interaction in the
polysome. The main model predicts gene transcription levels, the auxiliary
model specifies RNA half-lives. The two models can be pipelined iteratively,
but this was not included in the design of this study.
The authors gave the results at the speed of RNA polymerase equal to 200
and 500 nt/s. Why were these values chosen? Do there exist similar results
with other speed values, e.g., 800 or 1200 nt/s?
Response. Predictions under rates of 800 and 1200 are not provided here.
Our tests suggest that the rate of 800 produces even closer agreement with
the experiment compared to the rate of 500, whereas under 1200 this
congruence is poor. A preliminary prediction of the model is that the
polymerase elongation rate is 500–800 nt/s.
Is it possible to obtain similar results for other species, e.g., for chicken and
yeast?
Response. The yeast mitochondrial genome is known to contain 19
promoters. Solving an inverse problem is very computationally consuming
under so many unknown parameters, which exceeds available hardware
resources. In chicken, mitochondrial transcription is initiated at a single
bidirectional promoter or two overlapping promoters. However, more
experimental evidence on gene expression is needed. Therefore, predictions
are not provided for these organisms.
Quality of written English: Acceptable.
Acknowledgements
The authors are thankful to Prof. Alexander Kuleshov for help in and support
of this study. We thank Leonid Rusin for substantial contribution in
discussion and the manuscript preparation. We thank Drs. Brian Cusack and
Page 17 of 18
Rick Scavetta of Science Craft (www.science-craft.com) for editing the
manuscript.
This work was supported by the International Science and Technology
Center (grant 3807) and the Ministry for Education and Science of the
Russian Federation (grants P2370 and 14.740.11.0624).
Received: 13 March 2012 Accepted: 12 July 2012
Published: 9 August 2012
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